Single-input, multiple output (SIMO) and multiple-input, multiple-output (MIMO) wireless architectures are now common for high speed wireless communication systems. By means of multiple antennas and multiple processing units, e.g., transmit and receive RF chains, the quality of communications between wireless devices can be increased via diversity and multiplexing techniques. The advantages of MIMO architectures have been described extensively for communications systems, but not for ranging and positioning systems.
In SIMO and MIMO systems, multiple antenna elements are spaced to form phased array structures in radar and positioning applications. MIMO systems have been considered for radar applications for better detection and characterization of target objects.
FIG. 1A shows a conventional single-input, multiple-output (SIMO) phased array radar structure 100. In this example, there is one transmitter T1 101 including an antenna, and N receivers R1 102 through RN 103, each also including an antenna. Typically, the receivers and their antennas are approximately collocated at a single site.
A signal s(t) 111 is transmitted. The signal s(t) 111 arrives at the receivers as α s(t−τ) 112-113, where α is a channel coefficient and τ is a delay time. The signal received at time t at the ith receiver ri can be modeled asri(t)=αs(t−τ)+ni(t), tε[0,T]  (1)for i=1, . . . , N, in the time interval 0, T where T is the duration of the transmitted signal s(t) 111. The noise ni(t) is a complex valued white Gaussian noise process with zero mean and a spectral density σi2.
FIG. 1B shows conventional phased array processing. All the received signals 110 through 120 are aggregated 125, and then fed to a correlator 130. The output of the correlator is entered into a time of arrival (TOA) estimation unit 150, which returns an estimate of the TOA {circumflex over (τ)} 160. The channel coefficients for the received signals are constant and the same.
In statistics, the well known Cramér-Rao bound (CRB) or Cramér-Rao lower bound (CRLB) expresses a lower bound on a variance of an estimator of a deterministic parameter. The CRLB for the variance of the estimated delay time {circumflex over (τ)} 160 is
                              Var          ⁢                      {                          τ              ^                        }                          ≥                  1                      γ            ⁢                                                          α                                            2                        ⁢                                          ∑                                  i                  =                  1                                N                            ⁢                              1                                  σ                  i                  2                                                                                        (        2        )            where γ={tilde over (E)}−Ê2/E, α is the channel coefficient, N is the number of antennas and receivers, and σi2 is the variance of noise at the receiver ri. Also,
            E      ^        =                                  ∫                      -            ∞                    ∞                ⁢                                            s              ′                        ⁡                          (              t              )                                ⁢                                    s              *                        ⁡                          (              t              )                                ⁢                      ⅆ            t                                      ,E is the energy of the transmitted signal s(t) 111, and {tilde over (E)} is the energy of the first derivative of the signal s(t). The derivative of the signal s(t) is denoted as s′(t), and the complex conjugate of s(t) is denoted as s*(t).
It is seen from the CRLB Equation (2) that a significantly fading signal path can result in a substantially large CRLB. For the case of known channel coefficients, the square root of the corresponding CRLB of the variance of the distance estimate {circumflex over (d)} is
                                                        Var              ⁢                              {                                  d                  ^                                }                                              ≥                      c                          2              ⁢              π              ⁢                              N                            ⁢              β              ⁢                              SNR                                                    ,                            (        3        )            where c is the speed of light, N is the number of antenna elements at the receiver, SNR is the signal to noise ratio, and β is the effective bandwidth of the signal s(t).